Casazza et al. [1]} defined and studied \(\mathcal {X}_d\) -Bessel sequences and \(\mathcal {X}_d\) -frames in Banach spaces. Later on Stoeva [2]} studied perturbation of \(\mathcal {X}_d\) -Bessel sequences, \(\mathcal {X}_d\) -frames, atomic decomposition and \(\mathcal {X}_d\) -Riesz bases in separable Banach spaces. We have generalized this concept and defined approximative \(\mathcal {X}_d\) -Bessel sequences and approximative \(\mathcal {X}_d\) -frames in Banach spaces. We begin this section with the following definitions:
